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A Stochastic Kriging Approach for the Minimization of Integrals

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A Stochastic Kriging Approach for the Minimization of Integrals Felipe Carraro A stochastic Kriging approach for the minimization of integrals Florianópolis 2017 Felipe Carraro A stochastic Kriging approach for the minimization of integrals Dissertation submitted to the Civil Engineering Department of the Fed- eral University of Santa Catarina as a partial requirement to obtain the Master’s degree. Federal University of Santa Catarina Department of Civil Engineering Graduation Program in Civil Engineering Advisor: Prof. Rafael Holdorf Lopez Co-advisor: Prof. Leandro Fleck Fadel Miguel Florianópolis 2017 Ficha de identificação da obra elaborada pelo autor, através do Programa de Geração Automática da Biblioteca Universitária da UFSC. Carraro, Felipe A stochastic Kriging approach for the minimization of integrals / Felipe Carraro ; orientador, Rafael Holdorf Lopez, coorientador, Leandro Fleck Fadel Miguel, 2017. 157 p. Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós Graduação em Engenharia Civil, Florianópolis, 2017. Inclui referências. 1. Engenharia Civil. 2. minimização de integrais. 3. otimização global. 4. metamodelos. 5. kriging. I. Lopez, Rafael Holdorf. II. Miguel, Leandro Fleck Fadel. III. Universidade Federal de Santa Catarina. Programa de Pós-Graduação em Engenharia Civil. IV. Título. Felipe Carraro A stochastic Kriging approach for the minimization of integrals Esta Dissertação foi julgada adequada para a obtenção do Título de MESTRE em Engenharia Civil e aprovada em sua forma final pelo Programa de Pós-Graduação em Engenharia Civil - PPGEC da Universidade Federal de Santa Catarina. Florianópolis, 05 de outubro de 2017: Glicério Trichês, Dr. Coordenador PPGEC Comissão Examinadora: Prof. Rafael Holdorf Lopez, Dr. (Orientador) Universidade Federal de Santa Catarina Prof. Breno Pinheiro Jacob, Dr. (Videoconferência) Universidade Federal do Rio de Janeiro Prof. Marcelo Krajnc Alves, Ph.D. Universidade Federal de Santa Catarina Prof. Wellison José de Santana Gomes, Dr. Universidade Federal de Santa Catarina Resumo Este estudo tem como objetivo propor um método eficiente baseado na metodologia Stochastic Kriging (SK) para a solução de problemas de minimização de integrais. O modelo SK é usado para criar uma aproxi- mação rápida para a função a ser minimizada. Além disso, estimativas de variância da simulação de Monte Carlo são usadas para auxiliar a otimização. O procedimento de otimização aplica o algoritmo Efficient Global Optimization com o critério de preenchimento Augmented Ex- pected Improvement. Observa-se que o alvo de variância influencia os resultados da otimização. Um alvo muito baixo faz com a otimização torne-se demasiadamente custosa, enquanto que um alvo muito alto pode estagnar a otimização. Desta maneira, uma seleção adaptativa do alvo de variância é proposta. De modo a verificar o desempenho do método proposto vários testes comparativos são conduzidos. O método é aplicado para diversas funções de referência da literatura submetidas a um ruído estocástico. Além disso, comparações são realizadas em relação a um algoritmo eficiente conhecido e ao uso de quadraturas para a avaliação da integral. Por fim, a abordagem proposta é também apli- cada a um problema de engenharia estrutural. Os resultados destacam o desempenho eficiente do método bem como a sua consistência ao longo de diversas execuções independentes. Palavras-chave: stochastic kriging. otimização global. minimização de integrais. Resumo expandido Introdução Atualmente, muitas áreas da engenharia como civil, mecânica, naval e aeroespacial usam técnicas de otimização como uma ferramenta para resolver problemas reais. Apesar do aumento da capacidade computa- cional disponível, os ambientes de engenharia continuam a evoluir a analisar sistemas mais complexos, que podem incluir mais detalhes, por exemplo, modelos mais refinados. A análise desse tipo de problema, em termos de otimização, pode tornar-se inviável uma vez que uma única análise do modelo pode ser bastante custosa computacionalmente. Pode ser necessário a avaliação das incertezas inerentes ao problema estudado. Neste caso, há ainda um custo adicional relacionado ao fato do modelo analisado ser estocástico. Objetivos Este estudo tem como objetivo propor um método eficiente baseado na metodologia Stochastic Kriging (SK) para a solução de problemas de minimização de integrais. Mais especificamente busca-se estudar técnicas de metamodelagem associadas a simulações de Monte Carlo. A partir destas técnicas, objetiva-se a implementação de algoritmos computacionais bem como a realização de testes para a verificação do desempenho obtido. Uma das metas deste estudo também é propor uma maneira de incluir o erro cometido pela simulação de Monte Carlo na construção do metamodelo. Mais ainda, aplicar o método proposto a um problema de engenharia, como por exemplo o de controle ótimo da resposta dinâmica de estruturas sujeitas a carregamentos transientes. Por fim, busca-se verificar a eficiência dos métodos estudados. Metodologia O modelo SK é usado para criar uma aproximação rápida para a função a ser minimizada. Além disso, estimativas de variância da simulação de Monte Carlo são usadas para auxiliar a otimização. O procedimento de otimização aplica o algoritmo Efficient Global Optimization com o critério de preenchimento Augmented Expected Improvement. No entanto, este critério é modificado para que seja possível a inserção de pontos no modelo utilizando-se alvos de variância diversos. Mais ainda, de forma a balancear as características de exploração e refinamento da resposta, o alvo é alterado pelo uso de uma função de decaimento exponencial. Esta função realiza a adaptação do alvo conforme as características do ponto que deve ser adicionado ao metamodelo. De modo a verificar o desempenho do método proposto, vários testes comparativos são conduzidos. Inicialmente, o método é aplicado para quatro funções de referência da literatura submetidas a um ruído esto- cástico. As funções possuem 1, 2, 6 e 10 dimensões. O ruído estocástico é aplicado multiplicativamente a função determinística por meio de uma variável aleatória normal com média unitária e desvio padrão arbitrado para cada problema. Além disso, comparações são realizadas entre o método proposto e um al- goritmo eficiente conhecido, chamado Globalized Bounded Nelder–Mead (GBNM). Ainda, verifica-se existe vantagem em realizar a avaliação da integral que descreve o problema de otimização utilizando-se quadraturas ao invés da técnica de simulação de Monte Carlo. Por fim, a abordagem proposta é também aplicada a um problema de engenharia estrutural. Neste problema o objetivo é minimizar o valor esperado da probabilidade de falha de um pórtico plano de 10 pavimentos sujeito a ação sísmica. Uma forma de aleatoriedade do problema advém do sismo, que é simulado utilizando-se um processo estocástico filtrado pelo espectro Kanai-Tajimi. A outra, vem incorporada nos parâmetros estruturais de massa, amortecimento e rigidez. Estes são descritos por variáveis aleatórias com uma média fixa e um certo coeficiente de variação. A solução do problema se dá com uso do método proposto sendo que a parte dinâmica do problema é solucionada a partir da equação de Lyapunov. Esta equação descreve o problema em sua formulação de espaço de estado. Resultados e discussão Observa-se que o alvo de variância influencia os resultados da otimização. Um alvo muito baixo faz com a otimização torne-se demasiadamente custosa, enquanto que um alvo muito alto pode estagnar a otimização. É esta situação que motiva o uso de um esquema adaptativo para a seleção do alvo de variância. A partir dos resultados da otimização dos problemas referência da literatura, observam-se reduções no custo computacional e na variabilidade dos resultados quando da utilização do alvo adaptativo em relação ao constante. O uso de quadraturas, em sua forma mais simples com a utilização do produto de regras unidimensionais, acresce demasiadamente o número de avaliações para se chegar a uma resposta similar ao obtido pelo método proposto. Com relação ao algoritmo GBNM utilizado para comparação, o método proposto se mostrou mais eficiente e mesmo alterando-se os parâmetros internos daquele algoritimo não foi possível obter um número de avaliações equivalente para a partir do problema de duas dimensões. Os resultados destacam o desempenho eficiente do método bem como a sua consistência ao longo de diversas execuções independentes. Considerações finais De maneira geral pode-se dizer que os objetivos do estudo foram atingi- dos. Conseguiu-se propor um método que incorpora o erro de Monte Carlo na criação do metamodelo. Além disso, utilizando-se o algoritmo EGO foi possível realizar o refino iterativo do metamodelo objetivando a localização do ótimo global. Questões como a parada inesperada do algoritmo ou o alto número de avalições que estavam relacionadas ao valor do alvo de variância foram solucionados por meio do esquema adaptativo. O método proposto obteve uma maior eficiência com relação ao número de chamadas da função objetivo além de consistentemente chegar mais próximo ao ótimo global. Palavras-chave: stochastic kriging. otimização global. minimização de integrais. Abstract This study aims at proposing an efficient method based on the Stochastic Kriging (SK) methodology for the solution of integral minimization problems. The SK metamodel is used to create a fast approximation for the function being minimized. Moreover, variance estimates from Monte Carlo simulation are used to aid the optimization. The minimization procedure employed the Efficient Global
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